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We study the properties of the rook complex ℛ of a polyomino 풫 seen as independence complex of a graph G, and the associated Stanley–Reisner ideal Iℛ. In particular, we characterize the polyominoes 풫 having a pure rook complex, and the ones whose Stanley–Reisner ideal has linear resolution. Furthermore, we prove that for a class of polyominoes the Castelnuovo–Mumford regularity of Iℛ coincides with the induced matching number of G. [ABSTRACT FROM AUTHOR]

By a classical result of Brodmann, the function depthR/I^t is asymptotically a constant, i.e. there is a number s such that depthR/I^t = depthR/I^s for t > s. One calls the smallest number s with this property the index of depth stability of I and denotes it by dstab(I). This invariant remains mysterious till now. The main result of this paper gives an explicit formula for dstab(I) when I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize dstab(I) explicitly. The formula expresses dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals. [ABSTRACT FROM AUTHOR]

Let S = K [ x 1 , ... , x n ] , where K is a field, and t i (S / I) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i, where I is an edge ideal. In this paper, we prove that if t b (S / I) ≥ ⌈ 3 b 2 ⌉ for some b ≥ 0 , then the subadditivity condition t a + b (S / I) ≤ t a (S / I) + t b (S / I) holds for all a ≥ 0 . In addition, we prove that t a + 4 (S / I) ≤ t a (S / I) + t 4 (S / I) for all a ≥ 0 (the case b = 0 , 1 , 2 , 3 is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be an integer. The Grimaldi graph G(n) is defined by taking the elements of the set { 0 , … , n - 1 } as vertices. Two distinct vertices x and y are adjacent in G(n) if and only if gcd (x + y , n) = 1 . In this paper, we examine the Betti numbers of the edge ideals of these graphs and their complements. [ABSTRACT FROM AUTHOR]

Assume that G is a graph with edge ideal I(G). We provide sharp lower bounds for the depth of I (G) 2 in terms of the star packing number of G. [ABSTRACT FROM AUTHOR]

Assume that G is a graph with edge ideal I (G). For every integer s ≥ 1 , we denote the s -th symbolic power of I (G) by I (G) (s). It is shown that for every very well-covered graph G (without isolated vertices) and for each integer s ≥ 1 , we have depth S / I (G) (s) ≥ 2 c (G) − p (G) , where c (G) and p (G) denote the number of connected components and the number of bipartite connected components of G , respectively. Moreover, for any graph G , we determine a sharp lower bound for the depth of I (G) (3) in terms of the star packing number of G. [ABSTRACT FROM AUTHOR]

We study chains of nonzero edge ideals that are invariant under the action of the monoid Inc of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of Inc -invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of Inc -invariant chains of edge ideals. [ABSTRACT FROM AUTHOR]

Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, depth R / I ≥ 1 and depth R / I t = 0 for t ≫ 1 . This paper studies the problem when depth R / I t = 1 for some t ≥ 1 and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for depth R / I (t) = 1 for t ≫ 1 and show that the condition depth R / I (t) = 1 is persistent, where I (t) denotes the t-th symbolic powers of I. [ABSTRACT FROM AUTHOR]

Let ≥ 2 be an integer. The graph G n ¯ is obtained by letting all the elements of {0, ... , − 1} to be the vertices and defining distinct vertices and to be adjacent if and only if gcd(+ ,) ≠ 1. In this paper, we give some bounds for the Castelnuovo–Mumford regularity of the edge ideals and their powers for G n ¯ . [ABSTRACT FROM AUTHOR]

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king's graphs, and circular king's graphs. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly give combinatorial formulas for the regularity of powers of edge ideals, reg (I (D) k) , of weighted oriented unmixed forests D whose leaves are sinks ( V + (D) are sinks). This combinatorial formula is a piecewise linear function of k, for k ≥ 1 . [ABSTRACT FROM AUTHOR]

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement. [ABSTRACT FROM AUTHOR]

This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect [h,d]-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given.

Let D be a weighted oriented graph and I (D) be its edge ideal. We provide one method to find all the minimal generators of I ⊆ C , where C is a maximal strong vertex cover of D and I ⊆ C is the intersections of irreducible ideals associated to the strong vertex covers contained in C. If D ′ is an induced digraph of D , under a certain condition on the strong vertex covers of D ′ and D , we show that I (D ′) (s) ≠ I (D ′) s for some s ≥ 2 implies I (D) (s) ≠ I (D) s . We provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of D such that at most one edge is oriented into each of its vertices and w (x) ≥ 2 if deg D (x) ≥ 2 for all x ∈ V (D). Finally, if D is a weighted rooted tree with the degree of root is 1 and w (x) ≥ 2 when deg D (x) ≥ 2 for all x ∈ V (D) , we show that I (D) (s) = I (D) s for all s ≥ 2. [ABSTRACT FROM AUTHOR]

Let H be a simple undirected graph. The family of all matchings of H forms a simplicial complex called the matching complex of H. Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of H is Cohen–Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen–Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let G be a unicyclic graph with a unique odd cycle and I = I (G) be its edge ideal. We compute the exact values of all symbolic defects of I using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal I. We classify the class of unicyclic graphs when some power of maximal ideal annihilates I (s) / I s for any fixed s. Also for those class of graphs, we compute the Hilbert function of the module I (s) / I s for all s. [ABSTRACT FROM AUTHOR]

This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect [h; d]-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given. [ABSTRACT FROM AUTHOR]

Let H be a simple undirected graph and G = L (H) be its line graph. Assume that Δ (G) denotes the clique complex of G. We present a complete characterization of those H for which Δ (G) is Cohen–Macaulay or Gorenstein. In addition, we show that if Δ (G) is pure, then it is Cohen–Macaulay if and only if it is shellable if and only if it is vertex decomposable. [ABSTRACT FROM AUTHOR]

Let S be a ring of polynomials in finitely many variables over a field. In this paper, we give lower bounds for depth and Stanley depth of modules of the type S/It for t = 1, where I is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered. [ABSTRACT FROM AUTHOR]

In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered.

Introduction ‎‎ The study of the minimal free resolution of homogenous ideals and their powers is an interesting and active area of research in commutative algebra. Two invariants which measure the complexity of the minimal free resolutions are the so-called “projective dimension” and “Castelnuovo-Mumford regularity” (or simply, regularity) of the given ideal. Projective dimension determines the length of the minimal free resolution, while regularity is defined in terms of the degree of the entries of the matrices defining the differentials of the resolution. The focus of this paper is on the regularity of powers of ideals. One of the main results in this area is obtained by Cutkosky, Herzog, Trung [7], and independently Kodiyalam [8]. They proved that for a homogenous ideal I in a polynomial ring, the regularity of powers of I is asymptotically linear. In other words, there exist integers a(I) and b(I) such that regIs=aIs+b(I) for every integer s≫0. It is known that a(I) is bounded above by the maximum degree of generators of I. Moreover, if I is generated in a single degree d, then aI=d. But in general, it is not so much known about b(I) even if I is monomial ideal. However, when I is a quadratic squarefree monomial ideal, Alilooee, Banerjee, Beyarslan and Ha [9] conjectured that bI≤regI-2. In fact, they conjectured that the inequality regIs≤2s+regI-2 holds for any integer s≥1, when I is quadratic squarefree monomial ideal. Recently, Benerjee and Nevo [10] proved this conjecture for s=2. In this paper, we provide an alternative proof for their result. While the proof in [10] is based on topological arguments and using the Hochster’s formula, our proof is purely algebraic. Material and methods To every simple graph G one associates a quadratic squarefree monomial ideal, called its edge ideal, whose generators are the quadratic squarefree monomials corresponding to the edges of G. This association is a strong tool in the study of squarefree monomial ideals, as one can use the combinatorial properties of G to obtain information about the algebraic and homological properties of it, s edge ideal. One of the main results for bounding the regularity of powers of edge ideals is obtained by Benerjee [1]. He proved that the regularity of the sth power of an edge ideal I(G) has an upper bound which is defined in terms of the regularity of its (s-1)th power and the regularity of the edge ideal of some graphs which are explicitly determined by the structure of the G. This result has an essential role in our proof. Results and discussion The main result of this paper states that for every graph G, with edge ideal I(G), we have regIG2≤reg(IG)+2. In order to prove this inequality, using the aforementioned result of Benerjee, we must prove that the regularity of certain colon ideals are at most regIG. To achieve this goal, we use a short exact sequence argument which allows us to estimate the regularity of the colon ideas in terms of the regularity of edge ideal of some graphs which are strictly smaller than G. Conclusion The following conclusions were drawn from this research. The conjectured inequality of Alilooee, Banerjee, Beyarslan and Ha [9] is true for the case of s=2. It is known that for every graph G with edge ideal I(G) and induced matching number ν(G), we have 2s+νG-1≤reg(IGs), for every integer s≥1. Thus, our result implies that if regIG=νG+1, then regIG2=νG+3. The short exact sequence argument is a common technique in the study of regularity of monomial ideals. So, it would be interesting if one can prove the above-mentioned conjecture, using this method, even in the case of s=3 .

For a simple bipartite graph G , we give an upper bound for the regularity of powers of the edge ideal $I(G)$ in terms of its vertex domination number. Consequently, we explicitly compute the regularity of powers of the edge ideal of a bipartite Kneser graph. Further, we compute the induced matching number of a bipartite Kneser graph. [ABSTRACT FROM AUTHOR]

There are two motivating questions in [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, arXiv:1006.1087v1] and [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, J. Pure Appl. Algebra, 215(10) (2011), 2473-2480] about Castelnuovo-Mumford regularity and vertex decomposability of simple graphs. In this paper, we give negative answers to the questions by providing two counterexamples. [ABSTRACT FROM AUTHOR]

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if I=IΔ is the Stanley-Reisner ideal of a simplicial complex Δ, then reg(I(n))δ(n−1)+ b for all n1, where δ=limn→∞ reg(I(n))/n, b=max{reg(IΓ)|Γ is a subcomplex of Δ with F(Γ)⊆ F(Δ)}, and F(Γ) and F(Δ) are the set of facets of Γ and Δ, respectively. This bound is sharp for any n. When I=I(G) is the edge ideal of a simple graph G, we obtain a general linear upper bound reg(I(n))2n+ord-match(G)−1, where ord-match(G) is the ordered matching number of G. [ABSTRACT FROM AUTHOR]

Let \operatorname {pd}(I(G)) and \operatorname {reg}(I(G)) respectively denote the projective dimension and the regularity of the edge ideal I(G) of a graph G. For any positive integer n, we determine all pairs (\operatorname {pd}(I(G)),\, \operatorname {reg}(I(G))) as G ranges over all connected bipartite graphs on n vertices. [ABSTRACT FROM AUTHOR]

Assume that G is a chordal graph with edge ideal I(G) and induced matching number ν(G). We denote the sth symbolic power of I(G) by I(G)(s). It is shown that reg(I(G)(s))=2s + ν(G) − 1 for every integer s ≥ 1. [ABSTRACT FROM AUTHOR]

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs.

In this paper, we deal with very well-covered graphs. We first describe the structure of these kinds of graphs based on the structure of Cohen–Macaulay very well-covered graphs. As an application, we analyze the structure of local cohomology of the residue rings by the edge ideals of very well-covered graphs. Also, we give different formulas of regularity and depth of these rings from known ones and we finally treat the CM t property. [ABSTRACT FROM AUTHOR]

Minimal free resolutions of a finitely generated module over a polynomial ring S = k [ x ] , with variables x = { x 1 , ... , x n } and a field k have been extensively studied. Almost all the results in the literature about minimal free resolutions give their Betti numbers, that is, the ranks of the free modules in the resolution at each degree. Several techniques have been developed to compute Betti numbers, making this a manageable problem in many cases. However, a description of the differentials in the resolution is rarely given, as this turns out to be a more difficult problem. The main purpose of this article is to give a criterion to check when a graded free complex of an S-module is exact. Unlike previous similar criteria, this one allows us to give a description of the differentials using the combinatorics of the S-module. The criterion is given in terms of the Betti numbers of the resolutions in each degree and the set of columns of the matrix representation of the differentials. In the last section, and with the aim of illustrating how to use the criterion, we apply it to one of the first better-understood cases, the edge ideal of the complete graph. However, this criterion can be used to give an explicit description of the differentials of a resolution of several monomial ideals such as the duplication of an ideal, the edge ideal of a cograph, etc. [ABSTRACT FROM AUTHOR]

Let D be a weighted oriented graph and I(D) be its edge ideal. If D contains an induced odd cycle of length 2 n + 1 , under certain condition, we show that I (D) (n + 1) ≠ I (D) n + 1 . We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideals of weighted oriented graphs having each edge in some induced odd cycle of it. We characterize the weighted naturally oriented unicyclic graphs with unique odd cycles and weighted naturally oriented even cycles for the equality of ordinary and symbolic powers of their edge ideals. Let D ′ be the weighted oriented graph obtained from D after replacing the weights of vertices with non-trivial weights which are sinks, by trivial weights. We show that the symbolic powers of I(D) and I (D ′) behave in a similar way. Finally, if D is any weighted oriented star graph, we prove that I (D) (s) = I (D) s for all s ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered. [ABSTRACT FROM AUTHOR]

Let P be a finite poset and suppose I is the Stanley–Reisner ideal of the chain complex of P. In this paper, we prove the inequality reg (I (s)) ≤ 2 s + rank (P) for every integer s where I (s) denotes the sth symbolic power of I. As a consequence, we deduce that if I(G) is the edge ideal of a graph G such that its complementary graph Gc is bipartite, then the regularity of I (G) (s) is either 2s or 2 s + 1. [ABSTRACT FROM AUTHOR]

This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative. [ABSTRACT FROM AUTHOR]

compute depth and Stanley depth of the quotient rings of the edge ideals associated with different classes of graphs. These classes include some lobster trees and unicyclic graphs. We show that the values of depth and Stanley depth are equal for the classes we considered. [ABSTRACT FROM AUTHOR]

In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph G. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal I(G). We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs.

Suppose that F is a finite field and R = M n (F) is the ring of n -square matrices over F. Here we characterize when the Cayley graph of the additive group of R with respect to the set of invertible elements of R , called the unitary Cayley graph of R , is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay. [ABSTRACT FROM AUTHOR]

A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. [ABSTRACT FROM AUTHOR]

Introduction The study of the minimal free resolution of homogenous ideals and their powers is an interesting and active area of research in commutative algebra. Two invariants which measure the complexity of the minimal free resolutions are the so-called “projective dimension” and “Castelnuovo-Mumford regularity” (or simply, regularity) of the given ideal. Projective dimension determines the length of the minimal free resolution, while regularity is defined in terms of the degree of the entries of the matrices defining the differentials of the resolution. The focus of this paper is on the regularity of powers of ideals. One of the main results in this area is obtained by Cutkosky, Herzog, Trung [7], and independently Kodiyalam [8]. They proved that for a homogenous ideal I in a polynomial ring, the regularity of powers of I is asymptotically linear. In other words, there exist integers a(I) and b(I) such that reg (IS) = a(I) s + b (I) for every integer s » 0. It is known that a(I) is bounded above by the maximum degree of generators of I. Moreover, if I is generated in a single degree d, then a (I) = d. But in general, it is not so much known about b(I) even if I is monomial ideal. However, when i is a quadratic squarefree monomial ideal, Alilooee, Banerjee, Beyarslan and Ha [9] conjectured that b(I) ≤ reg (I) - 2. In fact, they conjectured that the inequality reg(Is) ≤ 2s + reg(I) - 2 holds for any integer s ≥ 1, when I is quadratic squarefree monomial ideal. Recently, Benerjee and Nevo [10] proved this conjecture for s = 2. In this paper, we provide an alternative proof for their result. While the proof in [10 ] is based on topological arguments and using the Hochster’s formula, our proof is purely algebraic. Material and methods To every simple graph G one associates a quadratic squarefree monomial ideal, called its edge ideal, whose generators are the quadratic squarefree monomials corresponding to the edges of G. This association is a strong tool in the study of squarefree monomial ideals, as one can use the combinatorial properties of G to obtain information about the algebraic and homological properties of it, s edge ideal. One of the main results for bounding the regularity of powers of edge ideals is obtained by Benerjee [1]. He proved that the regularity of the sth power of an edge ideal I(G) has an upper bound which is defined in terms of the regularity of its (s-1)th power and the regularity of the edge ideal of some graphs which are explicitly determined by the structure of the G. This result has an essential role in our proof. Results and discussion The main result of this paper states that for every graph G, with edge ideal I(G), we have reg(I(G) 2)≤ reg(I(G))+2. In order to prove this inequality, using the aforementioned result of Benerjee, we must prove that the regularity of certain colon ideals are at most reg(I(G)). To achieve this goal, we use a short exact sequence argument which allows us to estimate the regularity of the colon ideas in terms of the regularity of edge ideal of some graphs which are strictly smaller than G. Conclusion The following conclusions were drawn from this research. The conjectured inequality of Alilooee, Banerjee, Beyarslan and Ha [9] is true for the case of s=2. It is known that for every graph G with edge ideal I(G), and induced matching number v(G), we have 2s + v(G) -1 ≤ reg (I(G)s), for every integer s ≥ 1. Thus, our result implies that if reg(I(G)) = v(G)+1, then reg(I(g)²) = v(G) +3. The short exact sequence argument is a common technique in the study of regularity of monomial ideals. So, it would be interesting if one can prove the above-mentioned conjecture, using this method, even in the case of s=3. [ABSTRACT FROM AUTHOR]

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph. [ABSTRACT FROM AUTHOR]

In this paper we will provide the exact formulas for the regularity and projective dimension of edge ideals of three types of vertex-weighted oriented m-partite graphs. We also give some examples to show that the restrictions on orientations of edges and weights of vertices can not be removed. [ABSTRACT FROM AUTHOR]

Let D be a weighted oriented graph with the underlying graph G and I(D), I(G) be the edge ideals corresponding to D and G respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph remains same even after adding certain kind of new edges to it. We also establish the relationship between the regularity of edge ideal of weighted oriented path and cycle with the regularity of edge ideal of their underlying graph when vertices of V + are sinks. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

A graded ideal I in K [ x 1 , ... , x n ] , where K is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree pd (I) - 2 , while it is not linear at the homological degree pd (I) - 1 , where pd (I) denotes the projective dimension of I. In this paper, we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of K . We also compute the nonlinear Betti numbers of these ideals. Finally, we show that for the edge ideal I of a graph G with almost maximal finite index, the ideal I s has a linear resolution for s ≥ 2 if and only if the complementary graph G ¯ does not contain induced cycles of length 4. [ABSTRACT FROM AUTHOR]

Let S be a polynomial ring over a field K and I ⊂ S a homogeneous ideal. Let h (S / I , λ) be the h -polynomial of S / I and s = deg ⁡ h (S / I , λ) the degree of h (S / I , λ). It follows that the inequality s − r ≤ d − e , where r = reg (S / I) , d = dim ⁡ S / I and e = depth (S / I) , is satisfied and, in addition, the equality s − r = d − e holds if and only if S / I has a unique extremal Betti number. We are interested in finding a natural class of finite simple graphs G for which S / I (G) , where I (G) is the edge ideal of G , satisfies s − r = d − e. Let a (S / I (G)) denote the a -invariant of S / I , i.e., a (S / I (G)) = s − d. One has a (S / I (G)) ≤ 0. In the present paper, by showing the fundamental fact that every Cameron–Walker graph G satisfies a (S / I (G)) = 0 , a classification of Cameron–Walker graphs G for which S / I (G) satisfies s − r = d − e will be exhibited. [ABSTRACT FROM AUTHOR]

Let G be a finite simple connected graph on [n] and [ R = K[x1, . . . , xn] the polynomial ring in n variables over a field K. The edge ideal of G is the ideal I(G) of R which is generated by those monomials xixj for which {i, j} is an edge of G. In the present paper, the possible tuples (n, depth(R/I(G)), reg(R/I(G)), dim R/I(G), deg h(R/I(G))), where deg h(R/I(G)) is the degree of the h-polynomial of R/I(G), arising from Cameron–Walker graphs on [n] will be completely determined. [ABSTRACT FROM AUTHOR]

We classify all Gorenstein claw-free graphs. Moreover, we provide a new way to construct a Gorenstein graph from another one. [ABSTRACT FROM AUTHOR]

Let D be a weighted oriented graph, whose underlying graph is G, and let I (D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is Kőnig, we characterize when I (D) is unmixed. If G has no 3- or 5-cycles, or G is Kőnig, we characterize when I (D) is Cohen–Macaulay. We prove that I (D) is unmixed if and only if I (D) is Cohen–Macaulay when G has girth greater than 7 or G is Kőnig and has no 4-cycles. [ABSTRACT FROM AUTHOR]